OFFSET
1,2
COMMENTS
Key-numbers of the pitches of a major seventh chord on a standard chromatic keyboard, with root = 0.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(4 + 3*x + 4*x^2 + x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).
a(n) = (6*n - 4 + (-1)^n + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.
E.g.f.: ((6*x - 3)*cosh(x) + (6*x - 5)*sinh(x) + sqrt(2)*cos(x + Pi/4) + 2)/2.
Sum_{n>=2} (-1)^n/a(n) = log(3)/8 + log(2+sqrt(3))/(2*sqrt(3)) - 5*sqrt(3)*Pi/72. - Amiram Eldar, Dec 30 2021
MATHEMATICA
Select[Range[0, 200], MemberQ[{0, 4, 7, 11}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 11, 12}, 100]
PROG
(Magma) [n : n in [0..150] | n mod 12 in [0, 4, 7, 11]]
(PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+4*x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)))
CROSSREFS
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Phrygian mode (E): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): this sequence
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): A319452
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Sep 16 2018
STATUS
approved